Let $f:(0,+\infty)\to\mathbb{R}$ be a concave and continuous function. I know that $f$ satisfies the following inequalities (for all $x,y\in(0,\infty)$):
(1) $f(\lambda x+(1-\lambda) y)\geq\lambda f(x)+(1-\lambda)f(y)\;\;;0<\lambda<1$
(2) $f(x+h)+f(x-h)-2f(x)\leq0 \;\;; h>0$
(3) $\frac{f(x+h_2)-f(x)}{{h_2}}-\frac{f(x)-f(x-h_1)}{{h_1}}\leq0\;\;; h_1,h_2>0$
(4) $\frac{f(x_4)-f(x_3)}{{x_4}-x_3}\leq\frac{f(x_2)-f(x_1)}{{x_2}-x_1}\;\;; x_4>x_3>x_2>x_1$
(5) $\frac{1}{2}(f(x_2)+f(x_1))\leq\frac{1}{x_2-x_1}\int_{x_1}^{x_2}f(u)du\leq f(\frac{x_2+x_1}{2})$ In addition, I know that (1) and (4) are equivalent moreover (2) implies (3) also (3) implies (4) after that (4) implies (5). Now my question is : How can I prove that (2) implies (3) and how about (4) implies (5)? thanks a lot.